Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics

In this paper we look at $k$-stroll, point-to-point orienteering, as well as the deadline TSP problem on graphs with bounded doubling dimension and bounded treewidth and present approximation schemes for them. Given a weighted graph $G=(V,E)$, start node $s\in V$, distances $d:E\rightarrow \mathbb{Q}^+$ and integer $k$. In the $k$-stroll problem the goal is to find a path starting at $s$ of minimum length that visits at least $k$ vertices. The dual problem to $k$-stroll is the rooted orienteering in which instead of $k$ we are given a budget $B$ and the goal is to find a walk of length at most $B$ starting at $s$ that visits as many vertices as possible. In the P2P orienteering we are given start and end nodes $s,t$ for the path. In the deadline TSP we are given a deadline $D(v)$ for each $v\in V$ and the goal is to find a walk starting at $s$ that visits as many vertices as possible before their deadline. The best approximation for rooted or P2P orienteering is $(2+\epsilon)$-approximation [12] and $O(\log n)$-approximation for deadline TSP [3]. There is no known approximation scheme for deadline TSP for any metric (not even trees). Our main result is the first approximation scheme for deadline TSP on metrics with bounded doubling dimension. To do so we first show if $G$ is a metric with doubling dimension $\kappa$ and aspect ratio $\Delta$, there is a $(1+\epsilon)$-approximation that runs in time $n^{O\left(\left(\log\Delta/\epsilon\right)^{2\kappa+1}\right)}$. We then extend these to obtain an approximation scheme for deadline TSP when the distances and deadlines are integer which runs in time $n^{O\left(\left(\log \Delta/\epsilon\right)^{2\kappa+2}\right)}$. For graphs with treewidth $\omega$ we show how to solve $k$-stroll and P2P orienteering exactly in polynomial time and a $(1+\epsilon)$-approximation for deadline TSP in time $n^{O((\omega\log\Delta/\epsilon)^2)}$.